Chapter 11, Problem 11.25E

(a)

To determine

To find out what percentof people have WAIS scores above $100$ and above $145$ and below $85$ .

Answer to Problem 11.25E

$50\%$ of people have WAIS scores above $100$ , $0.15\%$ of people have WAIS scores above $145$ and $16\%$ of people have WAIS scores below $85$ .

Explanation of Solution

In the question, it is given that the scale of scores is set separately for each age group and is approximately Normal with mean $100$ and standard deviation $15$ . We will use $68-95-99.7$ rule to solve the problem. As we know that the empirical rule, also known as the three-sigma rule or the $68-95-99.7$ rule, provides a quick estimate of the spread of data in a normal distribution given the mean and standard deviation. Specifically, the empirical rule states that for a normal distribution:

$68\%$ of the data will fall within one standard deviation of the mean.

$95\%$ of the data will fall within two standard deviations of the mean.

Almost all $99.7\%$ of the data will fall within three standard deviations of the mean.

Thus, to calculate the percent of people have WAIS scores above $100$ we have that $100$ is the mean of the data then we can say that about $50\%$ of people have WAIS scores above $100$ as mean is the center of the normal graph.

And to calculate the percent of people have WAIS scores above $145$ , we have,

First calculate the z -value as,

  $\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ =\frac{145-100}{15}\\ =3\end{array}$

So, it has three standard deviations above the mean so, using $68-95-99.7$ rule we have,

  $\begin{array}{l}=\frac{1-0.997}{2}\\ =0.0015\\ =0.15\%\end{array}$

Thus, $0.15\%$ of people have WAIS scores above $145$ .

And to calculate the percent of people have WAIS scores below $85$ , we have,

First calculate the z -value as,

  $\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ =\frac{85-100}{15}\\ =-1\end{array}$

So, it has one standard deviation below the mean so, using $68-95-99.7$ rule we have,

  $\begin{array}{l}=\frac{1-0.68}{2}\\ =0.16\\ =16\%\end{array}$

Thus, $16\%$ of people have WAIS scores below $85$ .

(b)

To determine

To find out what percent of adults would qualify for membership.

Answer to Problem 11.25E

  $2.5\%$ of adults would qualify for membership.

Explanation of Solution

In the question, it is given that the scale of scores is set separately for each age group and is approximately Normal with mean $100$ and standard deviation $15$ . We will use $68-95-99.7$ rule to solve the problem. As we know that the empirical rule, also known as the three-sigma rule or the $68-95-99.7$ rule, provides a quick estimate of the spread of data in a normal distribution given the mean and standard deviation. Specifically, the empirical rule states that for a normal distribution:

$68\%$ of the data will fall within one standard deviation of the mean.

$95\%$ of the data will fall within two standard deviations of the mean.

Almost all $99.7\%$ of the data will fall within three standard deviations of the mean.

Thus, to calculate the percent of adults would qualify for membershipfor which scores should be $130$ or higher, we have,

First calculate the z -value as,

  $\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ =\frac{130-100}{15}\\ =2\end{array}$

So, it has two standard deviations above the mean so, using $68-95-99.7$ rule we have,

  $\begin{array}{l}=\frac{1-0.95}{2}\\ =0.025\\ =2.5\%\end{array}$

Thus, $2.5\%$ of adults would qualify for membership.

(c)

To determine

To find out about what percent of adults are mentally retarded according to this criterion.

Answer to Problem 11.25E

  $2.5\%$ of adults are mentally retarded according to this criterion.

Explanation of Solution

In the question, it is given that the scale of scores is set separately for each age group and is approximately Normal with mean $100$ and standard deviation $15$ . We will use $68-95-99.7$ rule to solve the problem. As we know that the empirical rule, also known as the three-sigma rule or the $68-95-99.7$ rule, provides a quick estimate of the spread of data in a normal distribution given the mean and standard deviation. Specifically, the empirical rule states that for a normal distribution:

$68\%$ of the data will fall within one standard deviation of the mean.

$95\%$ of the data will fall within two standard deviations of the mean.

Almost all $99.7\%$ of the data will fall within three standard deviations of the mean.

Thus, to calculate the percent of adults are mentally retarded according to this criterion for which scores should be $70$ or lower, we have,

First calculate the z -value as,

  $\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ =\frac{70-100}{15}\\ =-2\end{array}$

So, it has two standard deviations below the mean so, using $68-95-99.7$ rule we have,

  $\begin{array}{l}=\frac{1-0.95}{2}\\ =0.025\\ =2.5\%\end{array}$

Thus, $2.5\%$ of adults are mentally retarded according to this criterion.